One aspect that requires further consideration to advance the field of immersed finite element methods, is the detection of small features in the geometry or the solution. The decoupling of the geometry description from the finite element mesh makes the consideration of adaptive discretizations natural to immersed formulations. In this regard, it is notable that grid refinements in the vicinity of domain boundaries are commonly considered in the \(hp\) -adaptive finite cell method [1]}, [2]}, [3]}, [4]}. Such refinements are not required along all boundaries, but mainly in regions where small solution features such as large gradients are expected. For efficiency purposes, we therefore consider the further advancement of error estimators and error-estimation-based adaptivity schemes tailored to immersed formulations as presented in [5]} a desired development. One complication in regard of a posteriori error estimates is that the bilinear form is only coercive on discrete spaces, and that the dependence of the bilinear form on the discretization (both on the mesh size and on the order) complicates saturation assumptions. Besides the accuracy considerations related to enriched approximation spaces along small geometrical features, stabilized methods also require a certain level of geometry resolution, which makes local adaptive schemes well suited for the application of these techniques to problems on (scan-based) porous domains, as the refinements reduce the ratio of elements that are cut. Since element aggregation strongly ties degrees of freedom on badly cut elements to well posed degrees of freedom, local refinement procedures can be beneficial in cases where a large portion (or in certain regions even the majority) of the active mesh consists of badly cut elements. This is similar for ghost-penalty stabilization terms on domains with (relative to the mesh) thin structures, which can potentially make the domain artificially stiff in such cases. Note that in these cases the choice of the parameters \(\eta ^*\)  (with aggregation; see Section REF ) and \(\tau \)  (with ghost-penalty stabilization; see Section REF ) is particularly critical, and that in many cases problems can be avoided by a adequate parameter choice. Nevertheless, the accurate approximation of problems with very thin features in an unfitted setting remains a difficult problem for any method, and deserves further attention. Schwarz preconditioners are not affected by the matter of mesh resolution, but it should be noted that this only pertains to the solving of the linear system. With a natural boundary condition on the unfitted boundary and a preconditioner to resolve the conditioning problems, a relatively low level of mesh resolvement can therefore lead to reasonable solutions. It should be mentioned that this can still result in artificial couplings as described in e.g., [6]}, [7]}, however. With essential boundary conditions on the unfitted boundary, such as with flow problems in porous media, the application of a preconditioner without an additional stabilization technique will lead to inaccurate approximations, as described in Section .
